欧几里得等距群中共轭的几何

Quasiflats with holes in reductive groups

We give a new proof of a theorem of Kleiner-Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the rank of the Euclidean space is not less than the rank of the target. A bound on the size of the neighborhood and on the number of flats is determined by the size of the quasi-isometry constants. Without using asymptotic cones, our proof focuses on the intrinsic geometry of symmetric spaces and Euclidean buildings by extending the proof of Eskin-Farb's quasifla…

Geometry and Dynamics of Discrete Isometry Groups of Higher Rank Symmetric Spaces

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of ``nonelementary'' discrete isometry groups of higher rank symmetric spaces. We obtain…

Quasi-Euclidean tilings over 2-dimensional Artin groups and their applications

We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflats in all two-dimensional Artin groups -- atomic sectors. This allows us to provide useful quasi-isometry invariants for such groups -- completions of atomic sectors, stable lines, and the intersection pattern of certain abelian subgroups. These are described combinatorially, in t…

Structures of Coincidence Symmetry Groups

The structure of the coincidence symmetry group of an arbitrary $n$-dimensional lattice in the $n$-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries which are elements of the orthogonal group). Conditions under which the coincidence isometry group can be generated by reflections defined by vectors of the lattice will be discussed, and an algorithm to decompose an arbitrary element of the coincidence isometry group in terms of reflections defined by …

Defining relations for reflections. I

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and relations closely related to geometry. They allow to make fast and efficient geometric calculations. In this paper simple presentations of the isometry groups of Euclidean plane, 2-sphere, the real projective plane and groups SO(3), O(n) are introduced.

欧几里得等距变换的局部紧群的共轭长度

Conjugator length of locally compact groups of Euclidean isometries

研究欧几里得\(n\)维空间全等距群中特定局部紧子群\(H\)，通过分析其球面部分拓扑结构，证明其共轭长度函数有零增长、线性增长或无界三种情况，还指出仿射考克斯特群等共轭长度函数线性增长。

Reproducing kernel Hilbert spaces of CR functions for the Euclidean Motion group

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the Euclidean Motion $SE(2)$. A natural Hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable the resulting Hilbert space will not coincide with $L^2(SE(2))$. The reproducing kernel Hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms…

Quasiflats in CAT(0) complexes

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.

欧几里得建筑中余紧格的边界作用的稳定性

Stability for boundary actions of cocompact lattices in Euclidean buildings

研究欧几里得建筑中余紧格边界作用的稳定性，通过考虑等距离散群作用的扰动，证明足够小扰动与原始作用半共轭，得出半单\(p\) - 进李群中余紧格在\(G/Q\)上有拓扑稳定作用的结论。

The Coarse Geometry of Hartnell's Firefighter Problem on Infinite Graphs

In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the $n$-containment property if for every finite initial fire, there is a strategy to contain the fire by protecting $n$ vertices at each turn. A graph has the constant containment property if there is an integer $n$ such that it has the $n$-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. This result provides a unified way to recover previo…

Optimal higher-dimensional Dehn functions for some CAT(0) lattices

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $Γ$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $Γ$, namely the $k$-dimensional Dehn functions, which measure the difficulty to fill $k$-spheres by $(k+1)$-balls (for $1\leq k\leq \dim\ X-1$). Since the group $Γ$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $Γ$ are equivalent tothe corresponding results on fil…